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<article class="li"><h3 class="heading">
<span class="type">Item</span><span class="period">.</span>
</h3>
<p><dfn class="terminology">Integration in Time Domain</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}\left[\int_{0}^t f(\tau) d\tau \right]=\frac{F(s)}{s}.
\end{equation*}
</div>
<p class="continuation">This can be proven by realizing that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
f(t)*u_0(t)=\int_{0}^t f(\tau) u_0(t-\tau) d\tau    
=\int_{0}^t f(\tau) d\tau
\end{equation*}
</div>
<p class="continuation">and therefore by convolution property we have</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}[f(t)*u_0(t)]=F(s)\frac{1}{s}   .
\end{equation*}
</div>
<p class="continuation">Note <span class="process-math">\(u_0(t)=1\)</span> and <span class="process-math">\({\cal L}[u_0(t)]=1/s\text{.}\)</span></p></article><span class="incontext"><a href="sec8_2.html#li-72" class="internal">in-context</a></span>
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